AQA C2 (Core Mathematics 2) 2012 June

1 The arithmetic series $$23 + 32 + 41 + 50 + \ldots + 2534$$ has 280 terms.

1. Write down the common difference of the series.
2. Find the 100th term of the series.
3. Find the sum of the 280 terms of the series.

2 The triangle \(A B C\), shown in the diagram, is such that \(A B = 26 \mathrm {~cm}\) and \(B C = 31.5 \mathrm {~cm}\). The acute angle \(A B C\) is \(\theta\), where \(\sin \theta = \frac { 5 } { 13 }\).

1. Calculate the area of triangle \(A B C\).
2. Find the exact value of \(\cos \theta\).
3. Calculate the length of \(A C\).

3

1. \(\quad\) Expand \(\left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 }\).
2. Hence find \(\int \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x\).
3. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x\).

4 The \(n\)th term of a geometric series is \(u _ { n }\), where \(u _ { n } = 48 \left( \frac { 1 } { 4 } \right) ^ { n }\).

1. Find the value of \(u _ { 1 }\) and the value of \(u _ { 2 }\).
2. Find the value of the common ratio of the series.
3. Find the sum to infinity of the series.
4. Find the value of \(\sum _ { n = 4 } ^ { \infty } u _ { n }\).

5 The diagram shows a sector \(O P Q\) of a circle with centre \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-3_305_531_1105_758} The radius of the circle is 18 m and the angle \(P O Q\) is \(\frac { 2 \pi } { 3 }\) radians.

1. Find the length of the arc \(P Q\), giving your answer as a multiple of \(\pi\).
2. The tangents to the circle at the points \(P\) and \(Q\) meet at the point \(T\), and the angles \(T P O\) and \(T Q O\) are both right angles, as shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-3_597_529_1848_758}
   1. Angle \(P T Q = \alpha\) radians. Find \(\alpha\) in terms of \(\pi\).
   2. Find the area of the shaded region bounded by the \(\operatorname { arc } P Q\) and the tangents \(T P\) and \(T Q\), giving your answer to three significant figures.

6 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 4 } { x ^ { 2 } } - 11$$ The point \(P ( 2,1 )\) lies on the curve.

1. 1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 2\).
      (l mark)
   2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 2\).
   3. Hence state whether \(P\) is a maximum point or a minimum point, giving a reason for your answer.
2. Find the equation of the curve.

7 It is given that \(( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0\).

1. Find the possible values of \(\tan \theta\).
2. Hence solve the equation \(( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0\), giving all solutions for \(\theta\), in degrees, in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

8

1. Sketch the curve with equation \(y = 7 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
2. The curve \(C _ { 1 }\) has equation \(y = 7 ^ { x }\). The curve \(C _ { 2 }\) has equation \(y = 7 ^ { 2 x } - 12\).
   1. By forming and solving a quadratic equation, prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at exactly one point. State the \(y\)-coordinate of this point.
   2. Use logarithms to find the \(x\)-coordinate of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\), giving your answer to three significant figures.
      (2 marks)

9 The diagram shows part of a curve whose equation is \(y = \log _ { 10 } \left( x ^ { 2 } + 1 \right)\).
\includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-5_355_451_367_799}

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 0 } ^ { 1 } \log _ { 10 } \left( x ^ { 2 } + 1 \right) d x$$ giving your answer to three significant figures.
2. The graph of \(y = 2 \log _ { 10 } x\) can be transformed into the graph of \(y = 1 + 2 \log _ { 10 } x\) by means of a translation. Write down the vector of the translation.
3. 1. Show that \(\log _ { 10 } \left( 10 x ^ { 2 } \right) = 1 + 2 \log _ { 10 } x\).
   2. Show that the graph of \(y = 2 \log _ { 10 } x\) can also be transformed into the graph of \(y = 1 + 2 \log _ { 10 } x\) by means of a stretch, and describe the stretch.
   3. The curve with equation \(y = 1 + 2 \log _ { 10 } x\) intersects the curve \(y = \log _ { 10 } \left( x ^ { 2 } + 1 \right)\) at the point \(P\). Given that the \(x\)-coordinate of \(P\) is positive, find the gradient of the line \(O P\), where \(O\) is the origin. Give your answer in the form \(\log _ { 10 } \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.